On signpost systems and connected graphs

Download PDF
6 downloads 148 Views 274KB Size Report
Comment
uiSz and xySz. Then u0 . .... in [6], which is slightly different from that in [2]. ..... Since S is a simple smooth signpost system, Lemma 6 implies that u0 ...uiSz. By.
Czechoslovak Mathematical Journal

Ladislav Nebeský On signpost systems and connected graphs Czechoslovak Mathematical Journal, Vol. 55 (2005), No. 2, 283–293

Persistent URL: http://dml.cz/dmlcz/127978

Terms of use: © Institute of Mathematics AS CR, 2005 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz

Czechoslovak Mathematical Journal, 55 (130) (2005), 283–293

ON SIGNPOST SYSTEMS AND CONNECTED GRAPHS

   

   

, Praha

(Received May 20, 2002)

Abstract. By a signpost system we mean an ordered pair (W, P ), where W is a finite nonempty set, P ⊆ W × W × W and the following statements hold: if (u, v, w) ∈ P, then (v, u, u) ∈ P and (v, u, w) 6∈ P, for all u, v, w ∈ W ;

if u 6= v, then there exists r ∈ W such that (u, r, v) ∈ P, for all u, v ∈ W. We say that a signpost system (W, P ) is smooth if the folowing statement holds for all u, v, x, y, z ∈ W : if (u, v, x), (u, v, z), (x, y, z) ∈ P , then (u, v, y) ∈ P . We say thay a signpost system (W, P ) is simple if the following statement holds for all u, v, x, y ∈ W : if (u, v, x), (x, y, v) ∈ P , then (u, v, y), (x, y, u) ∈ P . By the underlying graph of a signpost system (W, P ) we mean the graph G with V (G) = W and such that the following statement holds for all distinct u, v ∈ W : u and v are adjacent in G if and only if (u, v, v) ∈ P . The main result of this paper is as follows: If G is a graph, then the following three statements are equivalent: G is connected; G is the underlying graph of a simple smooth signpost system; G is the underlying graph of a smooth signpost system. Keywords: connected graph, signpost system MSC 2000 : 05C38, 05C40, 05C12

By a graph we mean here a finite undirected graph with no multiple edges or loops. The letters f − n will serve for denoting non-negative integers. 1. Ternary systems By a ternary system we mean an ordered pair (W, P ) such that W is a finite nonempty set and P ⊆ W × W × W . Obviously, if (W, P ) is a ternary system, then P is a ternary relation in W . Research was supported by Grant Agency of the Czech Republic, grant No. 401/01/0218.

283

Let S = (W, P ) be a ternary system. Consider u0 , . . . , ui , v ∈ W , where i > 1. If (uj , uj+1 , v) ∈ P

for each j, 0 6 j 6 i − 1,

we will write u0 . . . ui Sv. Thus, instead of (u, v, w) ∈ P , where u, v, w ∈ W , we write uvSw. Moreover, instead of (x, y, z) 6∈ P , where x, y, z ∈ W , we will write ¬(xySz). Finally, we will write V (S) = W . We say that a ternary system S is smooth if it satisfies the following axiom (SMO): (SMO)

if uvSx, uvSz and xySz, then uvSy for all u, v, x, y, z ∈ V (S).

Lemma 1. Let S be a smooth ternary system and let u0 , . . . , ui , x, y, z ∈ V (S), where i > 1. Assume that u0 . . . ui Sx, u0 . . . ui Sz and xySz. Then u0 . . . ui Sy.



. We proceeed by induction on i. The case when i = 1 follows immediately from (SMO). Let i > 2. Obviously, u0 . . . ui−1 Sx and u0 . . . ui−1 Sz. By the induction hypothesis, u0 . . . ui−1 Sy. Recall that ui−1 ui Sx, ui−1 ui Sz and xySz. As follows from (SMO), ui−1 ui Sy. Thus u0 . . . ui Sy, which completes the proof.  Proposition 1. Let S be a smooth ternary system, and let u0 , . . . , ui , x0 , . . . , xj ∈ V (s), where i, j > 1. If (1)

u0 . . . ui Sx0 ,

u0 . . . ui Sz

and x0 . . . xj Sz,

then (2)

u0 . . . ui Sxj .



. Assume that (1) holds. We will prove that (2) holds. We proceed by induction on j. The case when j = 1 follows immediately from Lemma 1. Let j > 2. Obviously, x0 . . . xj−1 Sz. By the induction hypothesis, u0 . . . ui Sxj−1 . Since u0 . . . ui Sz and xj−1 xj Sz, Lemma 1 implies that u0 . . . ui Sxj , which completes the proof.  We say that a ternary system S is simple if it satisfies the following axioms (SIM1) and (SIM2): (SIM1) (SIM2)

if uvSx, xySv, then uvSy for all u, v, x, y ∈ V (S);

if uvSx, xySv, then xySu for all u, v, x, y ∈ V (S).

The next lemma depends on a result proved in [4].

284

Lemma 2. Let S be a simple ternary system, let w0 , . . . , wk , y, z ∈ V (S), where k > 1. Assume that w0 . . . wk Sy. If zySw0 , then w0 . . . wk Sz and zySwk . If yzSwk , then w0 . . . wk Sz and yzSw0 .



. The lemma immediately follows from Lemma 1 in [4].



Proposition 2. Let S be a simple ternary system, and let u0 , . . . ui , x0 , . . . , xj ∈ V (S), where i, j > 1. If (3)

ui . . . u0 Sx0

and x0 . . . xj Su0 ,

ui . . . u0 Sxj

and x0 . . . xj Sui .

then (4)



. Put h = i + j. Obviously, h > 2. Let (3) hold. We will prove that (4) holds. We proceeed by induction on h. First, let h = 2. Then i = 1 = j. Clearly, (SIM1) and (SIM2) imply (4). Let h > 3. If i = 1 or j = 1, then (4) immediatelly follows from Lemma 2. Assume that i, j > 2. Then h > 4. Since ui . . . u0 Sx0 and x0 . . . xj−1 Su0 , the induction hypothesis implies that ui . . . u0 Sxj−1 . Since ui−1 . . . u0 Sx0 and x0 . . . xj Su0 , the induction hypothesis implies that x0 . . . xj Sui−1 . It follows from (3) that ui ui−1 Sx0 and xj−1 xj Su0 . Since x0 . . . xj Sui−1 and ui ui−1 Sx0 , Lemma 2 implies that x0 . . . xj Sui . Since ui . . . u0 Sxj−1 and xj−1 xj Su0 , Lemma 2 implies that ui . . . u0 Sxj . Hence (4) holds. 

2. Signpost systems and their underlying graphs By a signpost system we mean a ternary system S satisfying the following axioms (SIG1), (SIG2) and (SIG3): (SIG1) if uvSw, then vuSu for all u, v, w ∈ V (S),

(SIG2) if uvSw, then ¬(vuSw) for all u, v, w ∈ V (S),

(SIG3) if u 6= v, then there exists r ∈ V (S) such that urSv for all u, v ∈ V (S). Remark. If S is a signpost system, u, v, w ∈ V (S) and uvSw, then the ordered triple (u, v, w) can be considered as a signpost showing a “direction” from u to w; this direction is determined by v. 285

The notion of a signpost system appeared in [2] but an implicit form of the idea of a signpost system can be found in [4] already. Our definition follows the definition in [6], which is slightly different from that in [2]. Note that (W, P ) is a signpost system (in the sense of our definition) if and only if P is a signpost system on W in the sense of the definition in [2]. Proposition 3. Let S be a signpost system. Then if uvSw, then uvSv for all u, v, w ∈ V (S),

(5)

uvSv if and only if vuSu for all u, v ∈ V (S)

(6) and (7)

if uvSw, then v 6= u 6= w for all u, v, w ∈ V (S).



. Obviously, (1) and (2) are implied by (SIG1). Combining (SIG1) and (SIG2) we get (3).  Lemma 3. Let S be a smooth signpost system. iom (SIM1).

Then S satisfies the ax-



. Consider arbitrary u, v, x, y ∈ V (S). Let uvSx, xySv. By (5), uvSv. If we put z = v in (SMO), we get uvSy.  Let S be a signpost system. By the underlying graph of S we mean the graph G defined as follows: V (G) = V (S) and u and v are adjacent in G if and only if uvSv for all u, v ∈ V (S).

Let S be a signpost system. By a confusing circuit in S we mean an ordered pair

(8)

(u0 u1 . . . uk , v)

such that u0 , u1 , . . . uk , v ∈ V (S), k > 1, u0 . . . uk Sv and uk = u0 .

286

Lemma 4. Let S be a signpost system, and let (8) be a confusing circuit in S. Then u0 6= v and k > 3.



. As follows from (7), u0 6= v and k > 2. Let k = 2. Since u2 = u0 , we get u1 u0 Sv and u0 u1 Sv, which contradicts (SIG2). Thus k > 3. 

Example. Let X, Y and Z be pairwise disjoint finite sets such that |X| > 3, |Y | > 1 and |Z| > 3. Put j = |X| and m = |Y |. Assume that X = {x0 , . . . , xj−1 } and Z = {z0 , . . . , zm−1 }. Moreover, put xj = x0 and zm = z0 . Let S be the ternary system defined as follows: V (S) = X ∪ Y ∪ Z and the following statement holds for all u, v, w ∈ V (S): uvSw if and only if one of the conditions (9)–(12) holds: u, v ∈ X ∪ Y

(9)

u, v ∈ Y ∪ Z

(10)

and u 6= v = w;

and u 6= v = w;

(11) there exists f, 0 6 f 6 j − 1, such that u = xf , v = xf +1 and w ∈ Z;

(12) there exists h, 0 6 h 6 m − 1, such that u = zh , v = zh+1 and w ∈ X. Obviously, S is a simple signpost system and its underlying graph is connected. It is clear that (x0 . . . xj−1 xj , z) and (z0 . . . zm−1 zm , x) are confusing circuits in S for every z ∈ Z and every x ∈ X. Let S be a simple signpost system. Assume that the underlying graph of S is connected. It was proved in [4] that there exists no confusing circuit in S if S satisfies the following axiom (ALT): (ALT)

if uvSx and xySy, then uvSy or xySu or yxSv for all u, v, x, y ∈ V (S).

The assumption that the underlying graph of a considered signpost system is connected is not needed in the next proposition: Proposition 4. Let S be a smooth signpost system. Then there exists no confusing circuit in S.



. Suppose, to the contrary, that there exist u0 , u1 , . . . , uk , v ∈ V (S) such that k > 1 and (8) is a confusing circuit in S. Then u0 u1 . . . uk Sv and u0 = uk . By Lemma 4, k > 3. Since u0 u1 Sv, Proposition 3 implies that u0 u1 Su1 . Since u0 = uk , Proposition 3 implies that ¬(u0 u1 Suk ). Then there exists j, 1 6 j 6 k − 1, such that u0 u1 Suj and ¬(u0 u1 Suj+1 ). Recall that u0 u1 Sv. Since S is smooth, we get ¬(uj uj+1 Sv), which contradicts the fact that u0 u1 . . . uk Sv. Thus the proposition is proved.  287

Corollary 1. Let S be a smooth signpost system, let u0 , . . . , ui , v ∈ V (S), where i > 1, and let u0 . . . ui Sv. Then (u0 , . . . , ui ) is a path in the underlying graph of S.



. Let G denote the underlying graph of S. Clearly, (u0 , . . . , ui ) is a walk in G. Suppose, to the contrary, that (u0 , . . . , ui ) is not a path. Then there exist j and k such that 0 6 j < k 6 i and uj = uk . Then (uj . . . uk , v) is a confusing circuit in S, which is a contradiction.  Let S be a smooth signpost system. By a path in S we mean a sequence (u0 , . . . , uk ), where k > 1, u0 , . . . , uk ∈ V (S) and u0 . . . uk Suk . As follows from Corollary 1, every path in S is a nontrivial path in the underlying graph of S. Let v0 , . . . , vm , u, v, w ∈ V (S), where m > 1; if (v0 , . . . , vm ) is a path in S, u = v0 , v = v1 and w = vm , then we say that (v0 , . . . , vm ) is an uv − w path in S. Theorem 1. Let S be a smooth signpost system, and let u, v, w ∈ V (S). If uvSw, then there exists an uv − w path in S.



. Suppose, to the contrary, that there exist no uv − w path in S. Then v 6= w. As follows from (SIG3), there exists an infinite sequence v0 , v1 , v2 , . . . of elements of V (S) such that v0 = u, v1 = v and v0 . . . vi Sw and vi 6= w for each i = 1, 2, 3, . . . . Since V (S) is finite, there exist distinct j and k such that 0 6 j < k and vj = vk . Since vj . . . vk Sw, we see that (vj . . . vk , w) is a confusing circuit in S, which contradicts Proposition 4. Thus the theorem is proved.  Remark. Theorem 1 can be interpreted as follows: Let S be a signpost system, let G denote the underlying graph of S, let u, v, w ∈ V (S) and uvSw. If we know that S is smooth, then we are certain that the signpost (u, v, w) shows a path going from u to w in G; the direction (in u) of this path is determined by v. The next lemma was proved in [4]: Lemma 5 (see Lemma 2 in [4]). Let S be a simple signpost system, and let u0 , . . . , ui−1 , ui ∈ V (S), where i > 1. If u0 . . . ui−1 ui Sui , then ui ui−1 . . . u0 Su0 . Hint for the proof. We proceed by induction on i. If i = 1, then we use (6). If i > 2, then we combine the induction hypothesis with Lemma 2 and (6). 

288

Corollary 2. Let S be a simple smooth signpost system, and let u0 , . . . , ui−1 , ui ∈ V (S), where i > 1. If (u0 , . . . , ui−1 , ui ) is a path in S, then (ui , ui−1 , . . . , u0 ) is a path in S.





. Proof is obvious.

3. Extensions of simple smooth signpost systems The next lemma will be used in the proof of the main result of this section. Lemma 6. Let S be a simple smooth signpost system, and let u0 , . . . , ui , v0 , . . . , vj , x, y ∈ V (S), where i, j > 1. Assume that u0 = v0 and ui = vj . If u0 . . . ui Sx and v0 . . . vj Sy,

(13) then

u0 . . . ui Sy and v0 . . . vj Sx.

(14)



. Let (13) hold. Since S is a smooth signpost system, Theorem 1 implies that there exist an ui−1 ui − x path in S and a vj−1 vj − y path in S. If ui = x, we put m = i. Let ui 6= x. Then there exist ui+1 , . . . um ∈ V (S), where m > i + 1, such that um = x and (ui−1 , ui , . . . , um ) is a path in S. If vj = y, we put n = j. Let vj 6= y. Then there exist vj+1 , . . . , vn ∈ V (S), where n > j + 1, such that vn = y and (vj−1 , vj , . . . , vn ) is a path in S. Clearly, (u0 , . . . , ui−1 , ui , . . . , um ) and (v0 , . . . , vj−1 , vj , . . . , vn ) are paths in S. Corollary 2 implies that (um , . . . , ui , ui−1 , . . . , u0 ) and (vn , . . . , vj , vj−1 , . . . , v0 ) are paths in S. Since u0 = v0 and ui = vj , we see that (vn , . . . , vj = ui , ui−1 , . . . , u0 )

and (um , . . . , ui = vj , vj−1 , . . . v0 )

are paths in S and therefore, by Corollary 2, (u0 , . . . , ui−1 , ui = vj , . . . , vn ) and (v0 , . . . , vj−1 , vj = ui , . . . , um ) are paths in S as well. Since vn = y and um = x, (14) follows, which completes the proof.  289

Corollary 3. Let S be a simple smooth signpost system, and let u0 , . . . , ui , x ∈ V (S), where i > 1. If u0 . . . ui Sx and u0 ui Sui , then u0 ui Sx.



. Proof is obvious.



A graph G is called a factor of a graph H if V (H) = V (G) and E(G) ⊆ E(H). Let S be a simple smooth signpost system, and let G denote the underlying graph of S. Consider a graph H such that G is a factor of H. By the extension of S to H we mean a ternary system S 0 with the following properties: (a) V (S 0 ) = V (S); (b) if u, v, w ∈ V (S 0 ), then uvS 0 w if and only if u and v are adjacent vertices of H and there exist u0 , . . . , ui ∈ V (S 0 ), where i > 1, such that u0 = u, ui = v and u0 . . . ui Sw. Clearly, the extension of S to H is defined uniquely. As follows from Corollary 3, the extension of a simple smooth signpost system S to the underlying graph of S is identical to S. Theorem 2. Let S be a simple smooth signpost system, and let G denote the underlying graph of S. Consider a graph H such that G is a factor of H. Then the extension of S to H is a simple smooth signpost system and H is its underlying graph.



. Put W = V (S). Let S 0 denote the extension of S to H. Obviously, S is a ternary system with V (S 0 ) = W . The proof of the theorem will be divided into four parts. We will prove that (1) S 0 is a signpost system, (2) H is the underlying graph of S 0 , (3) S 0 is smooth, and (4) S 0 is simple. Part 1. Consider arbitrary u, v, w ∈ W such that uvS 0 w. Then there exist u0 , . . . ui ∈ W , where i > 1, such that u0 = u, ui = v and u0 . . . ui Sw. Hence ui−1 ui Sw. Recall that S is a smooth signpost system. If ui = w, we put m = i. If ui 6= w, then, by Theorem 1, there exist ui+1 , . . . um ∈ W , where m > i, such that um = w and (ui−1 , ui , . . . , um ) is a path in S. Hence (u0 , . . . , ui , . . . , um ) is a path in S. By Corollary 2, (um , . . . , ui , . . . , u0 ) is a path in S as well. Hence ui . . . u0 Su0 . This means that vuS 0 u. We see that S 0 satisfies (SIG1). Assume that vuS 0 w. Then there exist v0 , . . . , vj ∈ W , where j > 1, such that v0 = v, vj = u and v0 . . . vj Sw. Since ui = v0 and vj = u0 , we see that (u0 . . . ui v1 . . . vj , w) is a confusing circuit in S, which contradicts Proposition 4. Hence ¬(vuS 0 w). We see that S 0 satisfies (SIG2). 0

290

Consider arbitrary u∗ , v ∗ ∈ W such that u∗ 6= v ∗ . Then there exists r ∈ W such that u∗ rSv ∗ . Since u∗ and r are adjacent vertices of H, we get u∗ rS 0 v ∗ . We see that S 0 satisfies (SIG3). Therefore, S 0 is a signpost system. Part 2. Obviously, V (H) = W . Consider arbitrary u, v ∈ W , u 6= v. As follows from the definition of S 0 , if uvS 0 v, then uv ∈ E(H). Conversely, let uv ∈ E(H). As follows from (SIG3), there exists r ∈ W such that urS 0 v. By Theorem 1, there exists an ur − v path in S. This implies that there exist u0 , . . . , uk , where k > 1, such that u0 = u, u1 = r, uk = v and u0 . . . uk Suk . Thus u0 uk S 0 uk ; we have uvS 0 v. Therefore, H is the underlying graph of S 0 . Part 3. Consider arbitrary u, v, x, y ∈ W such that uvS 0 x, uvS 0 z and xyS 0 z. Then uv, xy ∈ E(H) and there exist u0 , . . . , ui , v0 , . . . , vj , x0 , . . . , xk ∈ W , where i, j, k > 1, such that u0 = u, ui = v, v0 = u, . . . vj = v, x0 = x, xk = y, u0 . . . ui Sx, v0 . . . vj Sz

and x0 . . . xk Sz.

Since S is a simple smooth signpost system, Lemma 6 implies that u0 . . . ui Sz. By Proposition 1, u0 . . . ui Sxk . Recall that uv ∈ E(H). Hence uvS 0 y. We see that S 0 satisfies (SMO). Therefore, S 0 is smooth. Part 4. Consider arbitrary u, v, x, y ∈ W such that uvS 0 x and xyS 0 v. Then uv, xy ∈ E(H) and there exist u0 , . . . ui , x0 , . . . , xj ∈ W , where i, j > 1, such that u0 = v, ui = u, x0 = x, xj = y, ui . . . u0 Sxj and x0 . . . xj Su0 . Since S is simple, Proposition 2 implies that ui . . . u0 Sxj and x0 . . . xj Sui . Recall that uv, xy ∈ E(H). Hence uvS 0 y and xyS 0 u. We see that S 0 satisfies (SIM1) and (SIM2). Therefore, S 0 is simple. 

4. Connected graphs Let G be a connected graph. By the step system of G we mean the ternary system S such that V (S) = V (G) and uvSw if and only if d(u, v) = 1 and d(v, w) = d(u, w) − 1 for all u, v, w ∈ V (G), where d(x, y) denotes the distance between vertices x and y in G. It is easy to show that the step system of a connected graph is a simple signpost system. Lemma 7. If G is a tree, then the step system of G is smooth.



. Proof is simple.

 291

As was proved in [3] (and also in [5]), a simple signpost system is the step system of a connected graph if and only if (15)

the underlying graph of S is connected

and S satisfies the axiom (ALT) and a certain pair of additional axioms. Almost the same result was proved in [6]; but the proof in [6] is very different both from the proof in [3] and the proof in [5]; the pair of additional axioms was replaced by the following axiom (COM) in [6]: (COM)

if uvSx, vuSy and xySu, then yxSv for all u, v, x, y ∈ V (S).

The axioms (SIG1), (SIG2), (SIG3), (SIM1), (SIM2), (ALT) and (COM) (and, of course, the axiom (SWO)) could be formulated in a language of the first-order logic. It is unknown whether the condition (15) can be replaced by a finite set of axioms of the same kind. The step systems of connected graphs of two special classes were characterized without help of the condition (15) in [2]; one of these classes is the class of median graphs (for the notion of a median graph the reader is referred to [1]); note that every tree is a median graph. As was shown in [6], the step system of every median graph is smooth. On the other hand, if G is a connected graph that contains an induced K(2, 3), then the step system of G is not smooth. The following theorem is the main result of the present paper. Theorem 3. Let G be a graph. Then the following statements are equivalent: (a) G is connected; (b) G is the underlying graph of a simple smooth signpost system; (c) G is the underlying graph of a smooth signpost system.



. (a) → (b): Let G be connected. Consider an arbitrary spanning tree T of G. Let ST denote the step system of T . Then ST is a simple signpost system. By Lemma 7, ST is smooth. Let S denote the extension of ST to G. By Theorem 2, S is a simple smooth signpost system and the underlying graph of S is G. (b) → (c): Obvious. (c) → (a): Let G be the underlying graph of a smooth signpost system S. Consider arbitrary u, v ∈ V (S) such that u 6= v. By (SIG3), there exists t ∈ V (S) such that utSv. Theorem 1 implies that there exists an ut − v path in S. Clearly, there exists a path connecting u and v in G. This implies that G is connected.  292

References [1] H. M. Mulder: The Interval Function of a Graph. Math. Centre Tracts 132. Math. Centre, Amsterdam, 1980. [2] H. M. Mulder and L. Nebeský: Modular and median signpost systems and their underlying graphs. Discussiones Mathematicae Graph Theory 23 (2003), 309–324. [3] L. Nebeský: Geodesics and steps in a connected graph. Czechoslovak Math. J. 47(122) (1997), 149–161. [4] L. Nebeský: An axiomatic approach to metric properties of connected graphs. Czechoslovak Math. J. 50(125) (2000), 3–14. [5] L. Nebeský: A theorem for an axiomatic aproach to metric properties of graphs. Czechoslovak Math. J. 50(125) (2000), 121–133. [6] L. Nebeský: On properties of a graph that depend on its distance function. Czechoslovak Math. J. 54(129) (2004), 445–456. Author’s address: Univerzita Karlova v Praze, Filozofická fakulta, nám. J. Palacha 2, 116 38 Praha 1, Czech Republic, e-mail: [email protected].

293